The mathematical model of dynamical systems is represented as an initial-boundary value problem described by nonlinear vector-matrix valued partial differential equations. The linear partial differential operator associated with the nonlinear system is restricted to a time-invariant operator with domain dense in Hilbert space. New stability results reported in this paper show the existence of quadratic Lyapunov functions that yield both necessary and sufficient conditions for asymptotic stability of linear systems satisfying certain restrictions and the use of these forms for the stability investigation of a class of nonlinear systems. The proofs of the stability theorems employ the spectral representation of the Green’s function matrix of the associated linear differential operator. Therefore, in the earlier part of the paper well-known properties of linear operators are stated in order to express the Green’s function matrix in the form of spectral expansion.

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